Order of 2x2 Matrix [[2 3][3 5]]

[morningloree]

Hi!

I'm trying to find the order of a 2x2 matrix [[2 3][3 5]] (where the first row, from left to right, is 2 3 and the second row is 3 5). The definition of order that I am using is the least such power that I need to raise this matrix to in order to get the identity matrix. Also, I have to find the order in time proportional to the square root of the order, so I can't just do a linear search. Any help would be much appreciated!

 

[Jmf]

I'm not sure about an algorithm for solving this problem in general - but unless I misunderstood the problem then you're solving:

$$A^k = I$$

for A = [2,3;3,5] and k some integer? I'm pretty sure that a solution doesn't exist, as the values of the matrix will grow without bound as k goes large.